Differences of spectral projections and scattering matrix

The spectral density of a difference of spectral projections

Let $H_0$ and $H$ be a pair of self-adjoint operators satisfying some standard assumptions of scattering theory. It is known from previous work that if $λ$ belongs to the absolutely continuous spectrum of $H_0$ and $H$, then the difference of spectral projections $$D(λ)=1_{(-\infty,0)}(H-λ)-1_{(-\infty,0)}(H_0-λ)$$ in general is not compact and has non-trivial absolutely continuous spectrum. In this paper we consider the compact approximations $D_\varepsilon(λ)$ of $D(λ)$, given by $$D_\varepsilon(λ)=ψ_\varepsilon(H-λ)-ψ_\varepsilon(H_0-λ),$$ where $ψ_\varepsilon(x)=ψ(x/\varepsilon)$ and $ψ(x)…

具有算子系数的差分方程的散射理论

Scattering theory for difference equations with operator coefficients

本文研究具有算子系数的二阶差分方程，分析离散拉普拉斯算子的紧或迹类扰动，探讨其连续谱、离散谱及散射矩阵性质。

On the difference of spectral projections

For a semibounded self-adjoint operator $ T $ and a compact self-adjoint operator $ S $ acting on a complex separable Hilbert space of infinite dimension, we study the difference $ D(λ) := E_{(-\infty, λ)}(T+S) - E_{(-\infty, λ)}(T), \, λ\in \mathbb{R} $, of the spectral projections associated with the open interval $ (-\infty, λ) $. In the case when $ S $ is of rank one, we show that $ D(λ) $ is unitarily equivalent to a block diagonal operator $ Γ_λ \oplus 0 $, where $ Γ_λ $ is a bounded self-adjoint Hankel operator, for all $ λ\in \mathbb{R} $ except for at most countably many $ λ$. If, mor…

Spectral theory of discontinuous functions of self-adjoint operators and scattering theory

In the smooth scattering theory framework, we consider a pair of self-adjoint operators $H_0$, $H$ and discuss the spectral projections of these operators corresponding to the interval $(-\infty,λ)$. The purpose of the paper is to study the spectral properties of the difference $D(λ)$ of these spectral projections. We completely describe the absolutely continuous spectrum of the operator $D(λ)$ in terms of the eigenvalues of the scattering matrix $S(λ)$ for the operators $H_{0}$ and $H$. We also prove that the singular continuous spectrum of the operator $D(λ)$ is empty.

On Krein's Example

In his 1953 paper [Matem.~Sbornik \textbf{33} (1953), 597 -- 626] Mark Krein presented an example of a symmetric rank one perturbation of a self-adjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simple Lebesgue spectrum filling in the interval $[-1,1]$ and, therefore, the pair of the spectral projections is generic in the sense of Halmos but not Fredholm.

On a generalisation of Krein's example

We generalise a classical example given by Krein in 1953. We compute the difference of the resolvents and the difference of the spectral projections explicitly. We further give a full description of the unitary invariants, i.e., of the spectrum and the multiplicity. Moreover, we observe a link between the difference of the spectral projections and Hankel operators.